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A183158 T(n,k) is the number of partial isometries of an n-chain of fix k (fix of alpha is the number of fixed points of alpha). 4

%I

%S 1,1,1,4,2,1,12,6,3,1,38,10,6,4,1,90,26,10,10,5,1,220,42,15,20,15,6,1,

%T 460,106,21,35,35,21,7,1,1018,170,28,56,70,56,28,8,1,2022,426,36,84,

%U 126,126,84,36,9,1,4304,682,45,120,210,252,210,120,45,10,1

%N T(n,k) is the number of partial isometries of an n-chain of fix k (fix of alpha is the number of fixed points of alpha).

%H R. Kehinde, A. Umar, <a href="http://arxiv.org/abs/1101.0049">On the semigroup of partial isometries of a finite chain</a>, arXiv:1101.0049

%F T(n,0)= A183159(n). T(n,1)=A061547(n+1). T(n,k)=binomial(n,k), k > 1.

%e T (4,2) = 6 because there are exactly 6 partial isometries (on a 4-chain) of fix 2, namely: (1,2)-->(1,2); (2,3)-->(2,3); (3,4)-->(3,4); (1,3)-->(1,3); (2,4)-->(2,4); (1,4)-->(1,4) - the mappings are coordinate-wise.

%e ...1.

%e ...1....1.

%e ...4....2....1.

%e ..12....6....3....1.

%e ..38...10....6....4....1.

%e ..90...26...10...10....5....1.

%e .220...42...15...20...15....6....1.

%e .460..106...21...35...35...21....7....1.

%e 1018..170...28...56...70...56...28....8....1.

%e 2022..426...36...84..126..126...84...36....9....1.

%e 4304..682...45..120..210..252..210..120...45...10....1.

%p A183159 := proc(n) nh := floor(n/2) ; if type(n,'even') then 13*4^nh-12*nh^2-18*nh-10; else 25*4^nh-12*nh^2-30*nh-22; end if; %/3 ; end proc:

%p A061547 := proc(n) 3*2^n/8 +(-2)^n/24 - 2/3; end proc:

%p A183158 := proc(n,k) if k = 0 then A183159(n) ; elif k = 1 then A061547(n+1) ; else binomial(n,k) ; end if; end proc: # _R. J. Mathar_, Jan 06 2011

%t T[n_, 0] := (51*2^n + (-2)^n - 40)/12 - n*(n + 3);

%t T[n_, 1] := (9*2^n + (-1)^(n+1)*2^n - 8)/12;

%t T[n_, k_] := Binomial[n, k];

%t Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-Fran├žois Alcover_, Nov 22 2017 *)

%Y Cf. A183156 (row sums).

%K nonn,tabl

%O 0,4

%A _Abdullahi Umar_, Dec 28 2010

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Last modified January 24 04:35 EST 2020. Contains 331183 sequences. (Running on oeis4.)